In Asymptotic Statistics we study the asymptotic behaviour of (aspects of) statistical procedures. Here “asymptotic” means that we study limiting behaviour as the number of observations tends to infinity. A first important reason for doing this is that in many cases it is very hard, if not impossible to derive for instance exact distributions of test statistics for fixed sample sizes. Asymptotic results are often easier to obtain. These can then be used to construct tests, or confidence regions that approximately have the correct uncertainty level. Similarly, determining estimators or other procedures that are optimal in a specific sense, for instance in the sense of minimal mean squared error or variance, is often not possible if the number of observations is fixed. Using asymptotic results is it however in many cases possible to exhibit procedures that are asymptotically optimal.
In this course we begin by treating the mathematical machinery from probability theory that is necessary to formulate and prove the statements of asymptotic statistics. Important are the various notions of stochastic convergence and their relations, the law of large numbers and the central limit theorem (which the students are assumed to know), the multivariate normal distribution, and the so-called delta method. We will use these tools to study the asymptotic behaviour of statistical procedures.
It is assumed that students have at least successfully completed introductory courses on probability theory and statistics and courses on linear algebra and multivariate calculus. It is highly recommended to follow a course on measure theoretic probability.
Due to the Covid crisis the entire course will be online this year. Every week the schedule is as follows:
|1||9/9||introduction, convergence||slides+notes Sec. 1.1 up to and including Theorem 1.7||1.1, 1.2, 1.3(i), 1.4, 1.10, 1.15||watch hour 1 from minute 13:45||hour 1, hour 2|
|2||16/9||convergence||rest Sec. 1.1, Sec. 1.2||1.11, 1.12, 1.17, 1.28, 1.29, 1.32||1) Skip proof of Prohorov and Helly.|
2) There is an error in 1.28. Try to correct it!
|hour 1, hour 2|
|3||23/9||multivariate normal||Sec. 2.1-2.4||2.1, 2.2, 2.5, 2.8, 2.13, 2.16, 2.17||hour 1, hour 2|
|4||30/9||chi square test and delta method||Sec. 2.5, 3.1||2.22, 2.23(i), 3.1, 3.2, 3.3||Skip Theorem 2.10.||hour 1, hour 2|
|5||7/10||delta method||Sec. 3.2, 3.3||3.12, 3.18||watch hour 1 from 6:25||hour 1, hour 2|
|6||28/10||M-estimators, consistency||Chap. 4 up to and including p. 45 + extra material in slides||4.1, 4.5, 4.8, 4.11(i), 4.12(i, ii)||There is extra material about Glivenko-Cantelli theorems under bracketing in the slides||hour 1, hour 2|
|7||4/11||M-estimators, asymptotic normality, MLE||rest of Chap. 4||4.13, 4.14, 4.16, 4.18, 4.25||watch hour 1 from 23:55||hour 1, hour 2, hour 3|
|8||11/11||nonparametric estimation||Chapter 5||5.1, 5.2, 5.3, 5.6||hour 1, hour 2|
|18/11||live Q&A||Zoom link will be provided|
|9||25/11||minimax lower bounds 1||sections 6.1, 6.2||6.1||Material from the second set of lecture notes, see above||hour 1, hour 2, hour 3|
|10||2/12||minimax lower bounds 2||rest of chapter 6||6.2-6.7||hour 1, hour 2|
|11||9/12||high-dimensional models||chapter 7||7.3-7.5, 7.7-7.9||last lecture||hour 1, hour 2|